To make the cake: In a mixer set with the paddle attachment, beat the butter over medium speed until soft and creamy, about 5 minutes. Add the granulated sugar and continue to beat until well mixed and creamy, another 2 to 3 minutes. Beat in the egg yolks, add the lime zest and continue to beat until fully incorporated, another 2 minutes or so. Reduce the speed to low, and gently add the all-purpose flour, salt and baking powder. Continue to beat for another couple minutes. The batter should be completely homogeneous and feel a bit dense and heavy.
Transfer the batter to a large mixing bowl. Rinse the bowl of the mixer well and change the paddle attachment to the whisk attachment. Beat the egg whites on medium-high speed, until they hold stiff peaks, about 4 to 5 minutes. Slowly, incorporate a bit of the whipped-up egg whites into the heavy batter. Repeat with the rest of the egg whites, folding them gently but persistently, until the batter is completely incorporated and seems lighter and puffier. Don't overmix, or you will remove the puff effect from the beaten egg whites.
Preheat the oven to 350 degrees F. Butter the sides and bottom of a 14-cup bundt pan and lightly coat with all-purpose flour. Shake out any extra flour. Pour the batter into the prepared mold and bake in the oven about 1 hour and 10 minutes, or until the cake is golden brown on top, seems beautifully puffed up and a toothpick comes out moist but not wet. Remove from the oven. Let cool for at least 10 minutes.
To make the lime syrup: Combine the lime juice with the granulated sugar in a small saucepan and heat over medium heat for a couple minutes, just until the sugar has dissolved.
Run the tip of a knife around the edges of the cake. Place an inverted platter or board on top of the bundt pan. Invert to release the cake. With a toothpick or wooden skewer, poke holes all over its surface, going as deep as the toothpick will go--I poke at least 100 times! Pour half the lime syrup evenly over the cake. Let it absorb for a few minutes. Pour over the rest of the syrup and let it sit for a few minutes, while you make the glaze.
To make the lime glaze: In a small mixing bowl, using a small whisk or fork, combine the lime juice with the vanilla and confectioners' sugar until thoroughly mixed. With a tablespoon, add half the glaze, one tablespoon at a time, over the cake, slowly so it absorbs and coats the cake. Wait a few minutes and add the rest of the glaze. Let some of the extra glaze slide down the sides, as it will give it a pretty glazed look. Grate some fresh lime zest over the top.
If there is any leftover, cover so it will remain moist -- it usually becomes more moist after a day or two.